Quadratic Shear Research Articles

Soft polymers dynamic characterisation using sandwich theory

This paper presents a method to identify the dynamic mechanical properties of soft not self-supporting viscoelastic materials at high frequencies using sandwich beams. The motivation of this alternative method comes from the fact that the method of Ross, Kerwin and Ungar (RKU), proposed by the standard ASTM E 756-05, adds structural rigidity to the sandwich as shear increases, for example at high frequencies, and consequently, the stiffness of the core material is underestimated. The proposed method is based on a homogenization technique for sandwich beams that takes into account quadratic shear stress in the formulation, instead of a constant shear as the RKU method. To prove its worth, the properties of a soft polymer are identified by both methods from experimental transmissibility measures carried out on a sandwich beam up to 2500 Hz. Then, finite element simulations are performed using the obtained material properties to compare the numerical transmissibility functions with the experimental one. A numerical comparison for the case of a sandwich beam with a thicker viscoelastic core is also completed, proving the improvement in the stiffness and damping properties identification in the high frequency range by considering quadratic shear in the formulation. As a result, the identified mechanical properties by the proposed method are more accurate at high frequencies and for thicker core beams, i.e., when the shear effect is significant.

A new homogenization scheme for beam and plate structures without a priori requirements on boundary conditions

This contribution picks up on a novel approach for a first order homogenization procedure based on the Irving-Kirkwood theory and provides a finite element implementation as well as applications to beam and plate structures. It does not have the fundamental problems of dependency from representative volume element (RVE) size in determining the shear and torsional stiffness for beams and plates, that is present in classic Hill-Mandel methods. Due to the possibility of using minimal boundary conditions whilst simultaneously reusing existing homogenization algorithms, creation of models and numerical implementation are much more straight forward. The presented theory and FE formulation are limited to materially and geometrically linear problems. The approach to determining shear stiffness is based on the assumption of a quadratic shear stress distribution over the height (and width in case of the beam), which causes warping of the cross-section under transverse shear loading. Results for the homogenization scheme are shown for various beam and plate configurations and compared to values from well known analytical solutions or computed full scale models.

Unsteady Linearisation of Bed Shear Stress for Idealised Storm Surge Modelling

The modelling of time-varying shallow flows, such as tides and storm surges, is complicated by the nonlinear dependency of bed shear stress on flow speed. For tidal flows, Lorentz’s linearisation circumvents nonlinearity by specifying a (steady) friction coefficient r based on a tide-averaged criterion of energy equivalence. However, this approach is not suitable for phenomena with episodic and irregular forcings such as storm surges. Here, we studied the implications of applying Lorentz’s energy criterion in an instantaneous sense, so that an unsteady friction coefficient r(t) adjusts to the temporal development of natural wind-driven flows. This new bed-stress parametrisation was implemented in an idealised model of a single channel, forced by time-varying signals of wind stress (acting over the entire domain) and surface elevation (at the channel mouth). The solution method combines analytical solutions of the cross-sectionally averaged linearised shallow-water equations, obtained in the frequency domain, with an iterative procedure to determine r(t). Model results, compared with a reference finite-difference solution retaining the quadratic bed shear stress, show that this new approach accurately captures the qualitative and quantitative aspects of the surge dynamics (height and timing of surge peaks, sloshing, friction-induced tide-surge interaction) for both synthetic and realistic wind forcings.

Vibration characteristics of plates and shells with functionally graded pores imperfections using an enhanced finite shell element

The novelty of the current study is the development of an enhanced finite shell model using the first order shear deformation theory (FSDT) to study the vibration characteristics of imperfect functionally graded (FGM) plates and shells including porosities. Unlike standard (FSDT) theory, the proposed model presents an improvement of the FSDT via the introduction of a quadratic shear function allowing the exact representation of the parabolic distribution of the shear strains and avoiding the utilization of shear correction factors which are computationally costs. The finite element method is used for the discretization of the equations of motion and the assumed natural transverse shear strain (ANS) method is employed to avoid shear locking problems. The effect of porosity parameter and its distributions, namely even and uneven forms, is investigated through different numerical illustrative examples. The present finite shell element model provides better agreement with exact solutions in comparison by other theories. Finally, effects of different factors on vibrational behavior of imperfect FGM porous plates and shells are examined in detail.

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft2 on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.

A geometrically exact thin-walled beam element with simplified strain definition

This paper presents a study of the formulation of nonlinear beam elements for spatial thin-walled frame structures. A formulation based on the geometrically exact beam theory is developed first, which includes transverse shear deformation, torsion-related warping and all major nonlinear strain terms. A comprehensive investigation is then made on different terms in the nonlinear strain expressions and their effects on the solution accuracy in nonlinear stability and post-buckling analyses. It is found that some nonlinear terms in the strain expression for the Wagner effect, quadratic shear deformation and torsion-shear coupling may have a significant impact on the solution accuracy, and hence must be included in an element formulation. A simplified strain expression is proposed, and a new thin-walled beam element is developed for physically and geometrically nonlinear analyses by using the proposed expression and formulation. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed formulation and element.

Dynamic Analysis of Three-Layer Sandwich Beams with Thick Viscoelastic Damping Core for Finite Element Applications

This paper presents an analysis of the dynamic behaviour of constrained layer damping (CLD) beams with thick viscoelastic layer. A homogenised model for the flexural stiffness is formulated using Reddy-Bickford’s quadratic shear in each layer, and it is compared with Ross-Kerwin-Ungar (RKU) classical model, which considers a uniform shear deformation for the viscoelastic core. In order to analyse the efficiency of both models, a numerical application is accomplished and the provided results are compared with those of a 2D model using finite elements, which considers extensional and shear stress and longitudinal, transverse, and rotational inertias. The intermediate viscoelastic material is characterised by a fractional derivative model, with a frequency dependent complex modulus. Eigenvalues and eigenvectors are obtained from an iterative method avoiding the computational problems derived from the frequency dependence of the stiffness matrices. Also, frequency response functions are calculated. The results show that the new model provides better accuracy than the RKU one as the thickness of the core layer increases. In conclusion, a new model has been developed, being able to reproduce the mechanical behaviour of thick CLD beams, reducing storage needs and computational time compared with a 2D model, and improving the results from the RKU model.

Analysis of moderately thin-walled beam cross-sections by cubic isoparametric elements

In technical beam theory the six equilibrium states associated with homogeneous tension, bending, shear and torsion are treated as individual load cases. This enables the formulation of weak form equations governing the warping from shear and torsion. These weak form equations are solved numerically by introducing a cubic-linear two-dimensional isoparametric element. The cubic interpolation of this element accurately represents quadratic shear stress variations along cross-section walls, and thus moderately thin-walled cross-sections are effectively discretized by these elements. The ability of this element to represent curved geometries, and to accurately determine cross-section parameters and shear stress distributions is demonstrated.

Motion of a solid particle in a shear flow along a porous slab

AbstractThe flow field around a solid particle moving in a shear flow along a porous slab is obtained by solving the coupled Stokes–Darcy problem with the Beavers and Joseph slip boundary condition on the slab interfaces. The solution involves the Green’s function of this coupled problem, which is given here. It is shown that the classical boundary integral method using this Green’s function is inappropriate because of supplementary contributions due to the slip on the slab interfaces. An ‘indirect boundary integral method’ is therefore proposed, in which the unknown density on the particle surface is not the actual stress, but yet allows calculation of the force and torque on the particle. Various results are provided for the normalized force and torque, namely friction factors, on the particle. The cases of a sphere and an ellipsoid are considered. It is shown that the relationships between friction coefficients (torque due to rotation and force due to translation) that are classical for a no-slip plane do not apply here. This difference is exhibited. Finally, results for the velocity of a freely moving particle in a linear and a quadratic shear flow are presented, for both a sphere and an ellipsoid.

Faxén’s law for arbitrary oscillatory Stokes flow past a porous sphere

The present article deals with the study of the hydrodynamics of a porous sphere in an oscillatory viscous flow of an incompressible Newtonian fluid. Unsteady Stokes equations are used for the flow outside the porous sphere and Darcy’s equation is used for the flow inside the porous sphere. Corresponding Faxen’s law for drag and torque acting on the surface of the porous sphere is derived. Also the results are compared with few existing special cases. Examples like uniform flow, oscillating Stokeslet, oscillatory shear flow and quadratic shear flow are discussed.

Structural vibration of flexural beams with thick unconstrained layer damping

In this paper, the dynamic behaviour of free layer damping beams with thick viscoelastic layer is analysed. A homogenised model for the flexural stiffness is formulated employing Reddy and Bickford’s quadratic shear in each layer, in contrast to the classical model of Oberst and Frankenfeld for thin beams, which does not take into account shear deformations. The results provided by these two models in free and forced vibration are compared by means of finite element procedures with those of a 2D model, which considers extensional and shear stress, and longitudinal, transverse and rotational inertias. The viscoelastic material is characterised by a fractional derivative model, which takes into consideration the variation of the complex modulus with frequency. To avoid the frequency dependence of the stiffness matrices, the extraction of the eigenvalues and eigenvectors is completed by a new iterative method developed by the authors. The frequency response to a harmonic force is deduced by the superposition of modal contribution functions. From these numerical applications it can be concluded that the model for thick beams provides sufficient accuracy for practical applications, able to reproduce the mechanical behaviour of free layer damping beams with thick viscoelastic layer, reducing the storage needs and computational time with respect to a 2D model.

Adhesively bonded lap joints from pultruded GFRP profiles. Part III: Effects of chamfers

Quasi-static axial tension experiments were performed in a laboratory environment on epoxy bonded, balanced double-lap joints composed of pultruded GFRP flat profiles. The parameters investigated were the overlap length (from 50 to 100 mm), the adhesive layer thickness (1 and 3 mm) and the degree of chamfering of the adherends. The measured axial strain profiles in the joints correlated well with numerical results obtained from a 2D finite element analysis. Failure initiation and propagation always occurred in the outer fiber-mat layers of the adherends. Chamfering reduced and smoothed the through-thickness tensile and shear stress peaks towards the chamfered joint edges. However, joint strength was not significantly improved by chamfering. The joint strength was predicted by means of a quadratic shear–tensile failure criterion for a failure location in the outer fiber-mat layer of the pultruded adherends and compared well to the measurements. The adhesive layer thickness had an insignificant influence on the stress–strain distributions along the overlaps and joint strengh.

A diagrammatic analysis of quadratic shear viscous response

Abstract We study quadratic response in weakly coupled nonequilibrium φ 4 theory in the context of both classical transport theory and real time quantum field theory. A novel connection between these two approaches is established which provides a diagrammatic interpretation of quadratic shear viscous response.

Nonlinear response from transport theory and quantum field theory at finite temperature

We study nonlinear response in weakly coupled hot $\phi^4$ theory. We obtain an expression for a quadratic shear viscous response coefficient using two different formalisms: transport theory and response theory. The transport theory calculation is carried out by assuming a local equilibrium form for the distribution function and expanding in the gradient of the local four dimensional velocity field. By doing a gradient expansion on the Boltzmann equation we obtain a hierarchy of equations for the coefficients of this expansion.To do the response theory calculation we use Zubrave's techniques in nonequilibrium statistical mechanics to derive a generalized Kubo formula. Using this formula allows us to obtain the quadratic shear viscous response from the three-point retarded green function of the viscous shear stress tensor. We use the closed time path formalism of real time finite temperature field theoryto show that this three-point function can be calculated using equilibrium quantum field theory by writing it as an integral equation involving a four-point vertex. This four-point vertex can in turn be obtained from an integral equation which represents the resummation of an infinite series of ladder and extended-ladder diagrams.The connection between transport theory and response theory is made when we show that the integral equation for this four-point vertex has exactly the same form as the equation obtained from the Boltzmann equation for the coefficient of the quadratic term of the gradient expansion of the distribution function.

The behavior of closed inextensible membranes in linear and quadratic shear flows

The dynamics of a spheroidal vesicle, bounded by an inextensible membrane, is analyzed as a function of the enclosed fluid viscosity, and of the membrane mechanical properties. The two situations in which a bending rigidity and a shear elasticity are the potential energy source in the membrane have been compared, and the stability properties of quasi-spherical vesicle shapes discussed. The transition from tank-treading to flipping motion in an external shear flow, has been studied as a function of the vesicle internal viscosity and of the strength of the shear quadratic component. The transverse lift strength has been calculated in both cases of a tank-treading and of a flipping motion regime.

Simplified Model for Red Cell Dynamics in Small Blood Vessels

A simple mechanism for the confinement of red cells in the middle of narrow blood vessels is proposed. In the presence of a quadratic shear, red cells deform in such a way to loose fore-aft symmetry and to achieve a fixed orientation with respect to the flow. This leads to a drift away from the vessel walls, when the vessel diameter goes below a critical value depending on the viscoelastic properties and the dimensions of the cell.

Active Vibration Control of Intelligent Stiffened Laminates Using Smart Piezoelectric Materials by the Finite Element Method

The advent of high strength and light weight composite materials and newly developed smart materials (piezoelectric materials like polyvinylidene fluoride) have radically changed the concepts of advanced space structures and mechanical systems. Advanced structures with integrated self-monitoring and control capabilities are becoming very important due to the rapid development of intelligent mechanical systems and space structures. Since the structures are distributed and flexible in nature, distributed vibration measurement and active vibration suppression are of importance to their performance. A generalised finite element formulation of a consistent plate model for active vibration control of stiffened laminates integrated with piezoelectric polymer layers acting as distributed sensors and actuators is presented in this study. Total charge developed on the sensor layer is calculated from the direct piezoelectric equations. The equation of motion is discretized with a 9-noded Langrangian quadratic shear deformable isoparametric element. The piezoelectric sensor is not only distributed, but also integrated to the substrate since the output charge and so also the voltage is dependent on the sensor area. Also, the piezoelectric actuator induces the control moments at the ends of the actuators, Therefore, the number, size and location of the sensors and the actuators are very important in the control system design. By selective assembling of the element matrices for each electrode, responses of various sensor/actuator geometries can be investigated. For vibration control in the time domain, Newmark's direct integration method is used in the present investigation.

FREE VIBRATION ANALYSIS OF FIBER REINFORCED COMPOSITE BEAMS USING HIGHER ORDER THEORIES AND FINITE ELEMENT MODELLING

Three higher order refined displacement models are proposed for the free vibration analysis of sandwich and composite beam fabrications. These theories model the warping of the cross-section by taking the cubic variation of axial strain and they eliminate the need for a shear correction coefficient by assuming a quadratic shear strain variation across the depth of the cross-section. Numerical experiments for various lamination schemes, boundary conditions and aspect ratios are carried out to compare these models with the first order shear deformation theory, earlier investigations and also among themselves to ascertain the most efficient one. Numerical results for deep sandwich and composite beams are also presented for future references.

Finite-rotation theories for composite laminates

For the finite-element analysis of arbitrary composite laminates various finite-rotation theories are presented in a single formulation. First a refined theory is derived which allows a quadratic shear deformation distribution across the thickness. The so-called difference vector appearing in the kinematic relations is expressed in terms of rotational degrees of freedom permitting a clear determination of the deformed normal vector in every nonlinear range. The constitutive relations derived are applicable to orthotropic material properties varying arbitrarily across the thickness and to curvilinear laminate coordinates as well. This refined theory is then transformed into simplified theories of Kirchhoff-Love and Mindlin-Reissner types. Finally, the last formulation is used to formulate a layer-wise theory able to grasp the shear deformations very accurately. Kinematic relations and the corresponding constraints are presented in two alternative forms suitable for the application of isoparametric and classical finite-element formulations.

Refined shear-deformation models for composite laminates with finite rotations

For arbitrary multilayered shell structures made particularly of composite material layers a refined finite-rotation theory with seven independent displacement variables is developed, approximating the displacement field by a cubic series expansion of thickness coordinates. This model allows a quadratic shear deformation distribution across the thickness. Procedures are given permitting a unique determination of the first order displacement term in the case of finite rotations. Kinematic relations are formulated in two alternative forms suitable for both classical and isoparametric finite element formulations. The constitutive relations presented model orthotropic material properties varying arbitrarily across the thickness. This third order single-layer theory is then transformed, by introducing further constraints, into three simplified models : a third order theory with five independent displacement variables, a Mindlin-Reissner type theory and a Kirchhoff-Love type theory. These four models differ, however, from each other essentially in the constraints imposed on the first and third order displacement variables : a significant advantage for a unified finite element development. Finally, the Mindlin-Reissner type theory is generalized to a layer-wise model being the most predictive one in dealing with local interlaminar effects. The theoretical models are transformed into adequate finite shell elements and then compared by means of appropriate examples concerning their prediction capability. Also examples are given demonstrating their applicability to finite-rotation phenomena.